Introduction We are already familiar with operations of addition and subtraction of fractions. In this section let us discuss about the multiplication of fractions with the same denominator.

In the above diagrams, the first one shows the fraction $\frac{1}{3}$.

The second diagram shows $\frac{1}{3}$ of the shaded region of the first., which is $\frac{1}{3}$ of $\frac{1}{3}$

= $\frac{1}{3}\times \frac{1}{3}$

= $\frac{1}{9}$

The third diagram shows the $\frac{1}{3}$ of the previous shaded region, which is $\frac{1}{3}$ of $\frac{1}{9}$

= $\frac{1}{3}\times \frac{1}{9}$

= $ \frac{1}{27}$

Hence when we multiply the fractions with the same denominator and same numerator, we get the resulting fraction as the same part of each part.

**For example**:

$\frac{2}{5}$ of $ \frac{2}{5}$ is the same as $\frac{2}{5}\times\frac{2}{5}$

= $\frac{2\times 2}{5\times 5}$

= $\frac{4}{25}$

Hence when we find the same part of the part, we group the numerators, and denominators separately and multiply them separetly.

(i.e) $\frac{2}{5}\times \frac{2}{5}\times \frac{2}{5}$

= $\frac{2\times 2\times 2}{5\times 5\times 5}$

= $\frac{8}{125}$

We are aware that the product 2 x 2 x 2 x 2 x 2 can be expressed in exponent form as $2^{5}$

We use the same procedure for the fractions as well.

$\frac{2}{5}\times \frac{2}{5}\times \frac{2}{5}$ = $\left ( \frac{2}{5} \right )^{5}$

= $\frac{2^{5}}{5^{5}}$

= $\frac{32}{3125}$

In the above figure, first picture shows the fraction $\frac{2}{5}$

Where as the second picture, shows the $\frac{3}{5}$ of $\frac{2}{5}$ of the first picture.

Hence overall among 25 small boxes, 6 boxes are shaded, which is shown in the third picture.

Hence $\frac{3}{5}$ of $\frac{2}{5}$ is the same as $\frac{6}{25}$

For example : $\frac{3}{7}\times \frac{2}{7}\times \frac{4}{7}$ = $\frac{3\times 2\times 4}{7\times 7\times 7}$

=$\frac{24}{343}$

Hence we observe that when we find the product of the fractions with the same denominator, we get the smaller portion of the whole.

Arithmetically when we multiply the fractions with the same denominator, we follow the following steps.

**Step 1:**Group the numerators and multiply them.

**Step 2:**Group the denominators and multiply them.

**Step 3:**The resulting fraction from steps 1 and 2 is the product of the fractions with the same denominators.

**Step 4:**If the resulting fraction in step 3 is an improper fraction and has factors common, we divide the numerator and denominator by the Highest Common Factor and write the simplest form and convert into mixed fraction, else we highlight the obtained fraction as the final answer.

**Note**: When we multiply proper fractions with the same denominators, the resulting product will be the smallest of the whole. Hence it will definitely be a proper fraction.

#### Example 1: $\frac{5}{11}\times \frac{15}{11}\times \frac{20}{11}$

**Solution:**We have $\frac{4}{11}\times \frac{15}{11}\times \frac{20}{11}$

= $\frac{4\times 15\times 20}{11\times 11\times 11}$ [ grouping the numerators and the denominators separately ]

= $\frac{1200}{1331}$ [ multiplying the numbers in the numerator and denominator as grouped in the previous step ]

#### Example 2: $\frac{2}{9}\times \frac{4}{9}\times \frac{6}{9}$

**Solution:**We have $\frac{2}{9}\times \frac{4}{9}\times \frac{6}{9}$

=$\frac{2\times 4\times 6}{9\times 9\times 9}$ [ grouping the numerators and the denominators separately ]

= $\frac{48}{729}$ [ multiplying the numbers in the numerator and denominator as grouped in the previous step ]

=$\frac{48\div 3}{729\div 3}$ [ dividing the numerator and the denominator by the common factor 3 ]

= $\frac{16}{243}$ Final Answer in the simplest form.

Multiply the following fractions and express your answer in the simplest form.

1. $\frac{1}{6}\times \frac{1}{6}\times \frac{1}{6}$

2. $\frac{3}{10}\times \frac{7}{10}\times \frac{27}{10}$

3. $\frac{32}{12}\times\frac{24}{12} $

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